Re: insulation

Sounds like you need to insulate "inward." The closet should have room to install a "false" ceiling. I would use wall board. Your nailing grid would need to be 1x4's when using 3-1/2" fiberglass or 1x6's when using and if you prefer to use 6" batts.

I would install at 24" on center.

The R-value of wallboard is approx. R-6. Therefore, 3-1/2" fiberglass at R-13 and the wallboard gives you R-19, the old standard for overhead insulation. Of course, 6" fiberglass and the wallboard would give you R-25, closer to the standard of today which is R-30 in most areas. The fiberglass batt paper side is downward. It can be purchased in either 16" or 24" width.

Remember, with no heat ducts (usually) in closets, if you are trying to have your clothes warmer when putting them on in the morning, etc., then your chlices are to open the door to the closet when retiring, or install a louvered door.

Also, since your closet is on an outside wall, you may want to check to see if there is insulation in the "outside" sidewall of the closet. The recommended insulation of a "finished" sidewall is blown in cellulose or fiberglass. Materials and blowing machines are available a selected lumber yards.